A combinatorial calculation of the Landau-Ginzburg model $M= \mathbb C^3, W=z_1 z_2 z_3$

Abstract

The aim of this paper is to apply ideas from the study of Legendrian singularities to a specific example of interest within mirror symmetry. We calculate the Landau-Ginzburg $A$-model with $M= \mathbb C^3, W=z_1 z_2 z_3$ in its guise as microlocal sheaves along the natural singular Lagrangian thimble $L = {\mathit Cone}(T^2)\subset M$. The description we obtain is immediately equivalent to the $B$-model of the pair-of-pants $\mathbb P^1 \setminus \{0, 1, \infty\}$ as predicted by mirror symmetry.